1. Lecture 4 Mathematical and Statistical Models in Simulation

2. 2 Queueing Models  Simulation is often used in the analysis of queueing models.  Typical measures of system performance include server utilization (percentage of time server is busy), length of waiting lines, and delays of customers.  Decision maker is involved in trade-offs between server utilization and customer satisfaction in terms of line lengths and delays.  Simulation is often used in the analysis of queueing models.  Typical measures of system performance include server utilization (percentage of time server is busy), length of waiting lines, and delays of customers.  Decision maker is involved in trade-offs between server utilization and customer satisfaction in terms of line lengths and delays. Calling populationWaiting lineServer

3. 3  In a single-channel queueing system there are only two possible events that can affect the state of the system.  The entry of a unit into the system or the completion of service on a unit  The server has only two possible states:  it is either busy or idle  In a single-channel queueing system there are only two possible events that can affect the state of the system.  The entry of a unit into the system or the completion of service on a unit  The server has only two possible states:  it is either busy or idle Flow Diagram Departure event Being server idle time Remove the waiting unit from the queue Being servicing the unit Another unit waiting ? NO YES

4. 4 Example 2.1  Single-channel queue serves customers on a first-in, first-out (FIFO) basis Customer Number Arrival Time (Clock) Time Service Begins (Clock) Service Time (Duration) Time Service Ends (Clock) 10022 22213 36639 479211 59 112 615 419 Table 2.4. Simulation Table Emphasizing Clock Times

5. 5 Chronological Order  The occurrence of the two types of events Event TypeCustomer NumberClock Time Arrival10 Departure12 Arrival22 Departure23 Arrival36 47 Departure39 Arrival59 Departure411 Departure512 Arrival615 Departure619 Table 2.5. Chronological Ordering of Events

6. 6 Chronological Ordering (cont’)  Number in system at time t Number of customers in the system 1 2 0 48121620 123 4 4 5 56

7. 7Terminology  mean arrival rate (number of calling units per unit of time)   mean service rate of one server (number of calling units served per unit of time)  1/  mean service time for a calling unit  snumber of parallel service facilities in the system  L q mean length of the queue  Lmean number in the system (those in queue + being served)  W q mean time spent waiting in the queue  Wmean time spent in the system (W q + 1/  )   server utilization factor  mean arrival rate (number of calling units per unit of time)   mean service rate of one server (number of calling units served per unit of time)  1/  mean service time for a calling unit  snumber of parallel service facilities in the system  L q mean length of the queue  Lmean number in the system (those in queue + being served)  W q mean time spent waiting in the queue  Wmean time spent in the system (W q + 1/  )   server utilization factor

8. 8 Statistical Models in Simulation  Discrete Distribution – Poisson ( ) estimate “number of arrivals per unit time” where P(x) = the probability of X successes given a knowledge of = expected number of successes e = mathematical constant approximated by 2.71828 x = number of successes per unit  Discrete Distribution – Poisson ( ) estimate “number of arrivals per unit time” where P(x) = the probability of X successes given a knowledge of = expected number of successes e = mathematical constant approximated by 2.71828 x = number of successes per unit

9. 9 Poisson Distribution  Def: N(t) is a Possion process if  Arrivals occurs individually (at rate )  N(t) has stationary increments: The distribution of the numbers of arrivals between t and t+s depends on the length of the interval s and not on the starting point t.  N(t) has independent increments: The numbers of arrivals during nonoverlapping time intervals (t, t+s) and (t’, t’+s’) are independent random variables.  Def: N(t) is a Possion process if  Arrivals occurs individually (at rate )  N(t) has stationary increments: The distribution of the numbers of arrivals between t and t+s depends on the length of the interval s and not on the starting point t.  N(t) has independent increments: The numbers of arrivals during nonoverlapping time intervals (t, t+s) and (t’, t’+s’) are independent random variables.

10. 10 Uniform Distribution  Continuous Distribution – Uniform distribution A random variable x is uniformly distributed on the interval (a, b):  Continuous Distribution – Uniform distribution A random variable x is uniformly distributed on the interval (a, b):

11. 11 Uniform Distribution (cont’)  The uniform distribution plays a vital role in simulation. Random numbers, uniformly distribution between zero to 1, provide the means to generate random events.

12. 12 Exponential Distribution  Continuous Distribution – Exponential distribution has been used to model interarrival times when arrivals are completely random and to model service times which are highly variable. A random variable x is exponentially distributed with parameter >0:  Continuous Distribution – Exponential distribution has been used to model interarrival times when arrivals are completely random and to model service times which are highly variable. A random variable x is exponentially distributed with parameter >0:

13. 13Memoryless  Memoryless

14. 14 Example of Memoryless  Suppose that the life of an industrial lamp, in thousands of hours, is exponentially distributed with failure rate =1/3 (one failure every 3000 hours, on the average). Find the probability that the industrial lamp will last for another 1000 hours, given that it is operating after 2500 hours.